Fraction Calculator

Add, subtract, multiply and divide fractions — with the common denominator, the simplified answer and the mixed number, not just a decimal.

Reference table

First numerator	Result	As a mixed number	As a decimal
0.5	7/12	Already a proper fraction	0.583333
1	5/6	Already a proper fraction	0.833333
1.5	13/12	1 1/12	1.083333
2	4/3	1 1/3	1.333333
2.5	19/12	1 7/12	1.583333
3	11/6	1 5/6	1.833333
4	7/3	2 1/3	2.333333
5	17/6	2 5/6	2.833333
6	10/3	3 1/3	3.333333
7	23/6	3 5/6	3.833333
8	13/3	4 1/3	4.333333
9	29/6	4 5/6	4.833333
10	16/3	5 1/3	5.333333
12	19/3	6 1/3	6.333333
15	47/6	7 5/6	7.833333

What Actually Trips People Up

1/2 + 1/3 is not 2/5. You cannot add numerators and denominators separately — the halves and the thirds are different-sized pieces until you rewrite both over 6.
Dividing by a fraction makes the answer bigger. 1/2 ÷ 1/4 = 2, because a half contains two quarters.
Multiplying two proper fractions always makes the answer smaller. Two thirds of a half is a third, not a whole.
A tape measure is base 2: halves, quarters, eighths, sixteenths. Denominators there are always powers of two, which is why the common denominator is easy.

A Fraction Is a Division Waiting to Happen

Three quarters means three things divided among four, and the line between the numbers is a division sign that nobody bothered to rewrite. That single fact resolves most confusion about fractions. It explains why the denominator can never be zero — dividing by nothing has no answer. It explains why 8/4 is just 2. And it explains why a fraction with a numerator larger than its denominator, an improper fraction, is not improper in any mathematical sense: 7/4 is a perfectly valid number, one and three quarters, and the only thing improper about it is that older textbooks preferred the mixed form for reading aloud.

Why Adding Needs a Common Denominator

You can only add things that are the same size. Half a pizza and a third of a pizza are different-sized slices, so the count of slices tells you nothing until you cut both into the same size. Sixths work for both: a half is three sixths, a third is two sixths, and now the pieces match, so three plus two is five — five sixths. That is the entire method. Find a number both denominators divide into, rewrite each fraction over it by multiplying top and bottom by the same amount, then add or subtract the numerators only. Multiplication and division skip this step completely: you multiply straight across, and to divide you flip the second fraction and multiply.

The Six Mistakes

Adding across the top and bottom: 1/2 + 1/3 = 2/5. Wrong. The answer is 5/6, and 2/5 is smaller than either fraction you started with — a clear sign something broke.
Forgetting to multiply the numerator when rescaling: turning 1/2 into sixths gives 3/6, not 1/6. Whatever you do to the bottom, do to the top.
Dividing instead of flipping: 1/2 ÷ 1/4 is not 1/8. Flip the second fraction and multiply: 1/2 × 4/1 = 2.
Leaving the answer unsimplified. 6/8 is correct but 3/4 is the answer, and on an exam the difference costs marks.
Cancelling across a plus sign. You can cancel common factors when multiplying, never when adding.
Assuming a bigger denominator means a bigger number. 1/16 is smaller than 1/2 — more pieces means each one is tinier.

Where You Actually Meet Them

Recipes: halving ⅔ of a cup gives ⅓, but halving ¾ gives ⅜, which no measuring cup has — this is why recipes drop to tablespoons.
Tape measures and drill bits: everything is halves, quarters, eighths and sixteenths, so denominators are always powers of two.
Timber and pipe sizes: a 2×4 is not two by four inches, it is 1½ × 3½ after planing.
Music: a whole note, half note and quarter note are exactly the fractions they are named after.
Odds and probability: 1 in 6 for a die is the fraction 1/6, which is 16.67%.
Share prices used to be quoted in eighths of a dollar on US exchanges until decimalisation in 2001.

Worked Examples

Including the one everybody gets wrong

½ + ⅓

Common denominator of 2 and 3 is 6
1/2 becomes 3/6, and 1/3 becomes 2/6
3/6 + 2/6 = 5/6, already in lowest terms

5/6 — not 2/5

½ ÷ ¼

Flip the second fraction: 1/4 becomes 4/1
Multiply straight across: 1×4 = 4, 2×1 = 2
4/2 simplifies to 2

2 — a half holds two quarters

Frequently Asked Questions

Why is 1/2 + 1/3 not 2/5?

Because halves and thirds are different-sized pieces. Adding the numerators and denominators separately counts pieces of different sizes as if they were the same. Rewrite both over 6 — 3/6 and 2/6 — and the answer is 5/6. A quick sanity check: 2/5 is smaller than 1/2, and adding a positive number can never make the total smaller.

How do I find the common denominator?

Multiplying the two denominators always works: for 4 and 6 that gives 24. The lowest common denominator is smaller — 12 — and saves you simplifying later, but both give the same final answer once reduced.

How do I divide fractions?

Flip the second fraction upside down and multiply. 2/3 ÷ 1/6 becomes 2/3 × 6/1 = 12/3 = 4. Dividing by a fraction smaller than one always makes the answer bigger, which surprises people every time.

What is a mixed number?

A whole number written next to a proper fraction, like 1¾ instead of 7/4. It reads better out loud and matches how you'd measure something, but it is harder to calculate with — convert back to 7/4 before multiplying or dividing.

How do I simplify a fraction?

Divide the top and bottom by their greatest common divisor. For 18/24 that divisor is 6, giving 3/4. If the only number that divides both is 1, the fraction is already in lowest terms.

Can a denominator be zero?

No. The fraction bar is a division sign, and dividing by zero has no answer — not infinity, no answer at all. Any calculator that returns something for n/0 is lying to you.